A Low-Cost GPS/INS integration Methodology based on DGPMduring GPS outages
Yuexin Zhang, Lihui Wang, Nan Qiao, Xinhua Tang, Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of education, School of Instrument Science and Engineering, Southeast University, Nanjing, China
Bin Li, Beijing Research Center of Intelligent Equipment for Agriculture, Beijing, China
Abstract
How to achieve continuous, reliable and accurate positioning performance using low-cost sensors is one of the main challenges for aviation navigation system. Global Positioning System (GPS) can provide the primary means of navigation in a number of aviation navigation applications (e.g., manned and unmanned aircraft vehicle, airport ground vehicle). However, GPS signal deteriorations typically occur due to aircraft itself during maneuvering, ionospheric scintillation, Doppler shift, multipath and so on. Thus, there is a need to research GPS augmentation strategies which can be used in the Communication, Navigation, Surveillance/Air Traffic Management.GPS integration with Inertial Navigation system (INS) is one of the key strategies. But once GPS signal outages, the integrated navigation system works in pure INS, and positioning accuracy deteriorates with time. When using low cost GPS/INS integration, a primary problem is the rapid performance deteriorate during GPS outages. To provide continuous, accurate and reliable positioning information in aviation, discrete grey prediction model (DGPM) aided fusion methodology is proposed. The DGPM provides pseudo-GPS position information forINS during GPS outages. The mathematical model of integrated navigation system is established, including INS error equations, Kalman filter and DGPM. The model works in the update mode when there is no GPS failure, whereas it switches to the prediction mode in case of GPS outages. To verify the feasibility and effectiveness of the proposed methodology, real road test is performed. The comparison results show that accuracy of longitude and latitude are improved by more than 80% and 70%, respectively. The DGPM can effectively provide position corrections for standalone INS during GPS outages.
Introduction
GPS is a convenient and economical navigation and location technology, which has broad prospects in air traffic management, airport ground vehicle, precision landing and aviation flight[1][2]. But the GPS signal is lack of reliability and autonomy. To enhance the reliability of the navigation system, GPS/INS integrated navigation system has been useddue to its advantages of low cost and miniaturization[3][4]. However, if the vehicle is moving in complex environments such as under trees, with many tall buildings around, and in tunnel, etc., GPS may fail[5][6]. The integrated navigation system works in pure inertial navigation system. The system offer much lower levels of performance because of INS relatively larger biases and errors[7]. Thus, the primary concern when using INS is the rapid degradation in positioning performance during GPS outages.
There are two kinds of methods to solve this problem without adding additional sensors, one is to improve the combination mode, and the other is to improve the information fusion method. Compared with loosely coupled mode, when the available satellites is less than four, the tightly coupled mode has a certain damping effect on the divergence of navigation error[8]. However, the method is invalid when the GPS receiver has no output completely. Adding constraints to the integrated model can improve the navigation performance[9][10]. The scheme is simple, but the precision is limited and only applied to fixed paths. The neural network, genetic algorithm and support vector machine are used to train GPS information when GPS is available, the trained model can estimate the INS errors to improve the accuracy when GPS fails[11][12][13]. There limitation is much training data needed.
Compared with other time-series based predicting methods, the grey prediction requires only a limited amount of data to estimate the behavior of unknown system and is more robust with respect to noise and lack of modeling information [14][15]. However, the traditional grey prediction model is mainly applied to a single exponential growth data sequence, and it is often incapable of the fluctuation of the sequence data. Furthermore, the change of position components are uncertain. To overcome these shortcomings, the traditional model is improved to DGPM by using discrete form. The DGPM provides pseudo-GPS position information for INS during GPS outages. The mathematical model of integrated navigation system is established, including INS error equations, Kalman filter and DGPM. The model works in the update mode when there is no GPS failure, whereas it switches to the prediction mode in case of GPS outages.
To verify the feasibility and effectiveness of the proposed methodology, experimental tests were performed in the campus of Southeast University. A low-cost MEMS-IMU utilizing the Ellipse-A of SBG SYSTEMS sampled at 100Hz and TRIMBLE BD982 GNSS receiver with 1Hz rate were installed on vehicle. No natural GPS outages were detected, and thus the position information from this system can be used as a reference when evaluation the positioning accuracy. Three typical GPS outages (i.e., south-north movement, east-west movement and turning) were considered to evaluate the proposed methodology effectiveness. The experimental results showed that the DGPM can effectively provide position corrections for standalone INS during GPS outages. The proposed method can be used to enhance accuracy and continuity of aviation navigation data in conditions where GPS data is inaccurate/misleading or unavailable.
GPS/INS Integrated Navigation System
INS Error Equations
INS includes three-axis accelerometers and three-axis gyroscopes. According to the measured specific forces and angular increments, the navigation information (position, velocity and attitude) can be solved. The navigation frame adopts east-north-up (ENU) geographic coordinate system and the body frame adopts right-forward-up (RFU) coordinate system. The detailed derivation process of INS error equations can be referred to the literatures[16][17].
The attitude error of INS can be described as follows:
(1)
whereis attitude angle error under ENU frame, and are rotating angular velocity and its error were caused by the earth rotation respectively, and represent the angular velocity of the rotation of a navigation coordinate frame relative to earth and its error, is the gyroscope drifts onto the navigation coordinate frame.
The INS velocity error equation is given by
(2)
whereis the velocity error vector,is the specific force vector, denotes the velocity, indicates the biases of accelerometer onto the navigation frame.
The position error equation can be expressed as
(3)
where , and are latitude, longitude and height, respectively, , and are latitude error, longitude error and height error, respectively, and are the radii of the curvatures along the meridian and the radii of the curvatures along the parallel.
Kalman Filter
Considering the system precision and real-time, the fifteen dimensional state vector includes three attitude errors, three velocity errors, three position errors, three accelerometer biases and three gyroscope drifts. The state vector can be expressed as
(4)
where, and denote three accelerometer biases, respectively; , and denote three gyro drifts, respectively.
The state equation of the Kalman filter is
(5)
where is the state transition matrix of continuous system, which can be obtained by equations (1)-(3), is the state noise of continuous system.
The observable measurement of the Kalman filter is the difference between the position of INS and GPS. The measurement equation of the system is
(6)
where is the measurement matrix of continuous system, is the measurement noise of continuous system.
By discretization of the continuous system, the discrete state equation and the measurement equation can be obtained
(7)
where is the discrete system state transition matrix, is the system noise distribution matrix, and are state noise and measurement noise, represents the measurement matrix and is the measurement vector.
Generally, equations of the KF consist of the time predictionand measurement update[18][19]. The time prediction equations are responsible for the forward time transition of the current epoch states to the next epoch states and are given by
(8)
(9)
where is the priori estimate value of state variables, is the priori estimate error covariance, is the a posteriori estimate error covariance, and is system noise covariance.
The measurement update equations are given by
(10)
(11)
(12)
where is the Kalman filter, is the a posteriori estimate value of state variables, is the measurement error covariance.
Discrete Grey Prediction Model
Grey prediction theory regards the system as a function that varies with time. When modeling, it does not need a large amount of time series data and can achieve good predictive effect[20]. But the conventional grey prediction model lacks stability. For example, the continuous gray prediction model used in literature[15] usually only applies to pure exponential growth sequence, but the location of the vehicle does not necessarily meet this condition.The discrete grey prediction model solves the problem by the point view of from discrete to discrete. The DGPM (1,1) is established and the modeling steps are as follows.
(1) Data preprocessing
If there are negative numbers in the original data, the modeling condition (sequence level judgment) of DGPM cannot be satisfied. Usually, a constant can be added to the original data, and the original sequence is transformed into the nonnegative sequence. The sequence can be given as
(13)
whereis the length of the data series and can be determined according to the length of the assumed GPS outage time in the update mode.
(2) Accumulated generating operation
Take accumulated generating operation on , and the sequence can be obtained:
(14)
where .
(3) Establish DGPM (1, 1)
The DGPM (1, 1) model is established as:
(15)
where and are the development coefficients, , is the length of prediction sequence.
To solve the above equation, the coefficients and must be determined. They can be solved by means of the least squares method as follow
(16)
where , , .
If , then
(17)
(4) Inverse accumulated generating operation
Note that the DGPM (1, 1) model designed above is updated when GPS works normally. Once GPS fails, the latest DGPM(1,1) model is utilized to predict the corresponding position, which can be calculated by the inverse accumulated generating operation on . The estimated value is
(18)
The grey prediction value is the estimated value minus the constant added in the data preprocessing.
DGPM Aided GPS/INS Navigation
Figure 1 illustrates the structure of the DGPM aided GPS/INS integrated navigation system. The Fig. 1 (a) shows the updated mode when GPS is available., and are position, velocity and attitude, respectively, and indicates error. When GPS data is available, the whole system is in a loosely-coupled mode. The outputs of INS and GPS are integrated by KF, where the attitude, velocity, and position errors are estimated as a correction to the INS outputs. Once GPS fails, the system operation automatically switches to the prediction mode, as shown in Fig. 1 (b). In this case, the DGPM utilizes the latest GPS position to provide pseudo position , which is then used as the input of the KF to form the observation vector with the INS position. The navigation system will continuously give the integrated information during the GPS outages.
(a) The Updated Mode
(b) The Prediction Mode
Figure 1.DGPM Aided GPS/INS Integrated Navigation System Structure
Road Test Experiment and Results
To evaluate the performance of DGPM, experiment was conducted in the Jiulonghu campus of Southeast University in Nanjing. The INS consists of three accelerometers and three gyroscopes, where the gyroscope bias is and random walk is , the accelerometer bias in-run instability is and random walk is . The GPS real time kinematic (RTK) system includes a base station and a move station. The horizontal accuracy of GPS RTK can reach 8mm+1ppm, and the vertical accuracy is 15mm+1ppm. The update frequency of INS and GPS are 200 Hz and 10 Hz, respectively. The experimental photos are shown in Figure 2.
(a) The Base Station
(b) The Satellite Antenna and INS
Figure 2. The Experiment Photos
This trajectory was run for approximately 400s and three GPS outages were considered, which last for 20s. The selected GPS outages route included typical conditions such as south-north movement, east-west movement and turning. The Figure 3 shows the vehicle trajectory.
Figure 3. Route of the Vehicle (Courtesy of Google Earth)
Figure 4 and Figure 5 show the location of each of the 20 s GPS outages along the longitude and the latitude components, respectively. To briefly and directly show the prediction GPS position components results of DGPM, GPS 1st outage is chosen to show DGPM prediction errors, Taking the longitude and latitude of the GPS receiver output as an example, the five historical data before the current time are used to predict the current data. The predicted residuals for the longitude and latitude are shown in Fig 6. The predicted residual is less than 5 m, the DGPM prediction results can follow the dynamic changes of normal GPS data.
Figure 4. Longitude Component
Figure 5. Latitude Component
Figure 6. DGPM Prediction Errors during GPS 1st Outage
Figure 7 and Figure 8 illustrate the associated error distribution. From Fig.7 and Fig.8, it can be found that for each outage, the maximum position of pure INS is larger, which cannot meet the positioning requirements. It can be attributed to the fact that this method simply integrates GPS with low cost MEMS based IMU, which cannot ensure the poisoning accuracy and reliability for GPS outages. For example, the maximum longitude error for outage 2 reach 40.9m under pure INS mode, and with the DGPM aided, the maximum longitude error can reduce to 5.06m. It can be depicted that DGPM provides a better accuracy for both position component for all GPS outages.
Figure 7. Error for Longitude (3 outages)
Figure 8. Error for Latitude (3 outages)
A comparison to both pure INS and DGPM aided during GPS outages is performed for both the longitude and latitude position components during 20 s GPS outages. Table 1 gives their RMSE (root mean square error) values during three GPS outages, respectively. It can be observed from Table 1, the RMSEs of the proposed method are greatly reduced. For GPS 1stoutage, the accuracy of longitude and latitude are improved by 80.3% and 72.68%, respectively. For GPS 2ndoutage, the accuracy of longitude and latitude are improved by 80.83% and 74.33%, respectively. For GPS 3rdoutage, the accuracy of longitude and latitude are improved by 86.09% and 78.12%, respectively. It can be concluded that the proposed algorithm makes more stable and more accurate position than pure INS during GPS outages.
Table 1. RMSEs of Three GPS Outages (Unit: m)
Outage No. / Position Component / Pure INS / DGPM Aided1 / Longitude / 36.511 / 7.193
Latitude / 6.263 / 1.711
2 / Longitude / 17.093 / 3.277
Latitude / 10.001 / 2.567
3 / Longitude / 49.994 / 6.954
Latitude / 24.366 / 5.331
Conclusions
Discrete grey predict model is proposed to solve the positioning problems during the GPS outages. When the GPS signal is available, the integrated system works under GPS/INS loosely-coupled mode. And the DGPM stores the GPS position information. Once the GPS data fails, the well trained module will continuously provide pseudo-GPS position data to aid the pure INS. Field test data was collected to evaluate the performance of the DGPM. It canbeen seen that, during the 20 s GPS outages, the proposed DGPM is much better than the pure INS mode.
References
[1]Haltli B, Ewing P, Williams H, 2005, Global Navigation Satellite System (GNSS) and Area Navigation (RNAV) Benefiting General Aviation, Digital Avionics Systems Conference, pp. 13.A.5-1-13.A.5-8.
[2]Liu Ruihua, Zhao Yanan, WangJian, BaiJie, 2017, Research on Distortion of Space Signal of BeiDou Satellite Navigation System and the Influence of Civil Aviation Ranging Performance, Proceedings of the ION 2017 Pacific PNT Meeting, pp. 317-330
[3]Grewal M S, Weill L R, Andrews A P, 2001, Global Positioning Systems, Inertial Navigation, and Integration, Wiley Interdisciplinary Reviews Computational Statistics, pp. 383-384.
[4]Falco G, Pini M, Marucco G, 2017, Loose and Tight GNSS/INS Integrations: Comparison of Performance Assessed in Real Urban Scenarios, Sensors, pp. 255-282.
[5]Bowen Jiang, Kedong Wang, 2017, Algorithm Design for a Low-Cost Vehicle Integrated Navigation System in the GPS Outage, Chinese Journal of Sensors and Actuators, pp. 412-417.
[6]Costa E, 2011, Simulation of the Effects of Different Urban Environments on GPS Performance Using Digital Elevation Models and Building Databases, IEEE Transactions on Intelligent Transportation Systems, pp. 819-829.
[7]Brown A K, 2005, GPS/INS uses Low-cost MEMS IMU, Aerospace & Electronic Systems Magazine IEEE, pp. 3-10.
[8]Li X, Jiang R, Song X, et al, 2017, A Tightly Coupled Positioning Solution for Land Vehicles in Urban Canyons, Journal of Sensors, pp. 1-11.